Infinite volume Gibbs states of the generalized mean-field orthoplicial model

TL;DR

This paper rigorously classifies the infinite volume Gibbs states of a generalized mean-field orthoplicial model with continuous spins constrained on an $ ext{l}_1$-sphere, revealing they are convex combinations of product states.

Contribution

It introduces a probabilistic framework to classify Gibbs states of the model, employing large deviations, entropy, and ensemble equivalence techniques.

Findings

Gibbs states are convex combinations of product states.

The classification uses large deviation principles and entropy methods.

Exact integral representations are key technical tools.

Abstract

The generalized mean-field orthoplicial model is a mean-field model on a space of continuous spins on $\mathbb{R}^n$ that are constrained to a scaled $(n-1)$-dimensional $\ell_1$-sphere, equivalently a scaled $(n-1)$-dimensional orthoplex, and interact through a general interaction function. The finite volume Gibbs states of this model correspond to singular probability measures. In this paper, we use probabilistic methods to rigorously classify the infinite volume Gibbs states of this model, and we show that they are convex combinations of product states. The predominant methods utilize the theory of large deviations, relative entropy, and equivalence of ensembles, and the key technical tools utilize exact integral representations of certain partition functions and locally uniform estimates of expectations of certain local observables.